Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-25
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Left successor law for addition

Statement

For all m,n∈N: σ(m)+n=σ(m+n). (The recursion defines addition on the second argument; this is the matching law for adding on the first.)

Facts & Assumptions

Given: Addition with m+0=m and m+σ(k)=σ(m+k) (Addition of natural numbers).

[L1]

Induction principle (The principle of mathematical induction).

Proof

technique · induction
1.1

Base n=0: σ(m)+0=σ(m) by x+0=x, and σ(m+0)=σ(m) by m+0=m, so σ(m)+0=σ(m+0).

base
1.2

Inductive hypothesis: σ(m)+n=σ(m+n).

ih
2.1

Step: σ(m)+σ(n)=σ(σ(m)+n)=σ(σ(m+n))=σ(m+σ(n)), applying x+σ(k)=σ(x+k) at the outer sum, then the hypothesis, then m+σ(n)=σ(m+n) backward inside σ.

step 1.2given
3.1

By induction [L1], σ(m)+n=σ(m+n) for all n, and since m was arbitrary, for all m,n∈N.

step 1.1step 2.1discharge-induction∎

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources